Given an ODE of the form $$ay''(x)+by'(x)+cy(x)=f(x)$$
I know that we can find the general solution by finding the complementary function and particular solution and then adding them. My question is, why does this work?
Method of solving second order ODEs
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ordinary-differential-equations
1 Answers
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It works because this equation is linear in terms of $y$. To be precise, the differential operator $$P = a\frac{d^2}{dx^2} + b\frac{d}{dx} + cI,$$ acting on a space of twice differentiable functions is linear.
Therefore, if two functions $y_1$ and $y_2$ both solve the equation $$Py=f,$$then by taking the difference of their equations we get $$0 = f-f = Py_1 - Py_2=P(y_1-y_2),$$the latter equality being the critical part where the linearity of $P$ comes into play.
Therefore, all solutions of the initial differential equation differ by a solution of the corresponding homogenous equation. In other words, you can find a particular solution $y_p$, and then find a solution $y_h$ of the homogenous equation, then $y_p+y_h$ also solves the initial equation.